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Subject:   RE: exponents
Author: Mathman
Date: Oct 19 2004
On Oct 19 2004, Susan wrote:

My reply was general, Susan.

What I'm alluding to is the following, which we seem to agree upon, and which
amounts to what generates the idea in the first place, I think.  Here is how I
introduced it [below], and how they can see the continued pattern.  It is
"pattern" that must make sense to them for them to "see" it and to recall it.
However, that is always an individual task and problem for the ones seeing it
for the first time.
They can not have the insight and connections we have gained over time.  It is
brand spanking new.

As you say, the idea may be readily grasped, and then there seems to be
confusion in their minds at a later date.  I think that comes a large part from
lack of "practice".  We don't have or spend the time turning over the rocks; we
are in too much of a hurry to get through the woods.  Perhaps we forget how long
it took for us to really get ideas embedded in our own minds as we pursued our
own studies, applying them time and again in other areas, and the struggle at
first which now seems so easy to us as everything coalesced with our wider
experience.  I was perhaps fortunate to have my education when I did.  We used
logarithms and slide-rule, so the learning period was fairly constant and
fairly long.  If we read a book on calculus, through at one sitting, it should
all make sense because it does...but we will have learned nothing for the most
part, if we can comprehend it at all in the one sitting.  That is what takes the
time.  Each concept is new, and today there are too many concepts and too little
time.  I never saw coordinates until my graduating year [a LONG time ago.]  Then
we studied Cartesian geometry in depth.  It made sense, and was immediately
continued into the in-depth study.  Today coordinates are learned at a very
young age without application.  Then it is expected they will be recalled when
needed.  Mind, you had to be a serious student then as now.  It was like
learning the 6 trig functions.  That isn't trig, it's just the start, and it's
the "beginning" concepts that are the most difficult to grasp, no matter how
simple they become later on.
Re method: Consider the following.

Choose any base [here, 5]:

In the left column, keep decreasing by 1.
In the right column keep dividing by the base.
[Or add/multiply accordingly on the rise.]

.      .
.      .
.      .
5^4 = 625
5^3 = 125
5^2 = 25
5^1 = 5
5^0 = 1
5^-1 = 1/5
5^-2 = 1/25
5^-3 = 1/125
5^-4 = 1/625
.      .
.      .
.      .

It is a natural sequence with no new definition of meaning, just a recognition
of pattern.  This is so no matter the base chosen.  Numbers symmetrical about 1
are reciprocal.

Students can get lots of visual aid [practice] from making lists with other
bases to get the same results, either by hand [actually calculating values for
exponents and improving their "number sense"] or with more modern technology,
calculator or spreadsheet, which is supposed to free them to look into the
bigger picture.  Either way, the initial idea is simple, but like the game of
GO, that's just the start.

In any event, I do hope that what has been discussed here by all concerned is of
value to anyone listening in as well as to each other.  I am open to any
suggestion that I can pass along to colleagues hereabout, and will pop in now
and then with some points I think might be of interest and some use.

A final note:  There will always be some students who do not grasp the idea, as
there will always be some ideas we do not grasp ourselves.  Now and then they
surprise us, and sometimes teach us instead.  That's the struggle, the name of
the game.


>Thanks David, and again, I've used exactly the same
> procedures for teaching this concept.  I guess that the students
> understand why it works, but when it comes to looking at 2^-1 the
> "why" gets lost.  You are giving them the last step of the "why"
> presentation and expecting them to work backwards, or to remember
> what happened to get to that last step.  I've tried lots of ways to
> help them remember--for example, I have tried to rename the negative
> exponent as a "flipper" .  I think they get confused because -2
> means the opposite of 2, but 2^-1 means the reciprocal of 2, so the
> negative plays a different role in each context.  I try to emphasize
> this difference by giving them lots of practice, but I think it is
> confusing from the student perspective.

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